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A Large Scale Approach to Decomposition SpacesFeb 20 2019Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces, shearlet spaces ... More Twisting $c_0$ around nonseparable Banach spacesFeb 20 2019We consider the class of Banach space $Y$ for which $c_0$ admits a nontrivial twisted sum with $Y$. We present a characterization of such space $Y$ in terms of properties of the $weak^\ast$ topology on $Y^\ast$. We prove that under the continuum hypothesis ... More On the bi-Lipschitz geometry of lamplighter graphsFeb 19 2019In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most $6$. It follows that lamplighter graphs over countable ... More Pointwise Multipliers between weighted Copson and Cesàro function spacesFeb 19 2019In this paper the solution of the pointwise multiplier problem between weighted Copson function spaces $\operatorname{Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Ces\`{a}ro function spaces $\operatorname{Ces}_{p_2,q_2}(u_2,v_2)$ is presented, where $p_1,\,p_2,\,q_1,\,q_2 ... More Repeated quasi-integration on locally compact spacesFeb 19 2019When $X$ is locally compact, a quasi-integral (also called a quasi-linear functional) on $ C_c(X)$ is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, ... More Row contractions annihilated by interpolating vanishing idealsFeb 18 2019We study similarity classes of commuting row contractions annihilated by what we call higher order vanishing ideals of interpolating sequences. Our main result exhibits a Jordan-type direct sum decomposition for these row contractions. We illustrate how ... More Generalized Bessel and Frame MeasuresFeb 18 2019Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we introduce $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts ... More Constructing Subspace Packings from Other PackingsFeb 17 2019The desiderata when constructing collections of subspaces often include the algebraic constraint that the projections onto the subspaces yield a resolution of the identity like the projections onto lines spanned by vectors of an orthonormal basis (the ... More Metric currents and polylipschitz formsFeb 16 2019We construct, for a locally compact metric space $X$, a space of polylipschitz forms $\Gamma^*_c(X)$, which is a pre-dual for the space of metric currents of $\mathscr{D}_*(X)$ Ambrosio and Kirchheim. These polylipschitz forms may be seen as a substitute ... More Re-expansions on compact Lie groupsFeb 16 2019In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general ... More Antipodal Hadwiger numbers of finite-dimensional Banach spacesFeb 14 2019Let $X$ be a finite dimensional Banach space; we introduce and investigate a natural generalization of the concepts of Hadwiger number $H(X)$ and strict Hadwiger number $H'(X)$. More precisely, we define the antipodal Hadwiger number $H_\alpha(X)$ as ... More Spectral Action in Noncommutative GeometryFeb 14 2019What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry \`a la Connes, deliberately unveiling the answers to these questions. After a brief ... More Remarks on the strict order propertyFeb 14 2019A well-known theorem of Shelah asserts that a theory has $OP$ (the order property) if and only if it has $IP$ (the independence property) or $SOP$ (the strict order property). We give a mild strengthening of Shelah's theorem for classical logic and a ... More $L^1$-spaces of vector measures with vector densityFeb 13 2019Let $F$ be a function with values in a Banach space. When $F$ is locally (Pettis or Bochner) integrable with respect to a locally determined positive measure, a vector measure $\nu_F$ with density $F$ defined on a $\delta$-ring is obtained. We present ... More Embeddings of Orlicz-Lorentz spaces into $L_1$Feb 13 2019In this article, we show that Orlicz-Lorentz spaces $\ell^n_{M,a}$, $n\in\mathbb N$ with Orlicz function $M$ and weight sequence $a$ are uniformly isomorphic to subspaces of $L_1$ if the norm $\|\cdot\|_{M,a}$ satisfies certain Hardy-type inequalities. ... More Zero Jordan product determined Banach algebrasFeb 13 2019A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\varphi\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever $a$, $b\in A$ are such ... More Shadowing and structural stability in linear dynamical systemsFeb 12 2019A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by P. Hartman in 1960 for operators on finite-dimensional spaces. The ... More A Gleason-Kahane-Żelazko theorem for the Dirichlet spaceFeb 12 2019We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of ... More Direct and inverse limits of normed modulesFeb 11 2019The aim of this note is to study existence and main properties of direct and inverse limits in the category of normed $L^0$-modules (in the sense of Gigli) over a metric measure space. The Kato square root problem on irregular open setsFeb 11 2019Let $O\subseteq \mathbb{R}^d$ be an open set and $L=-\nabla \cdot A\nabla$ be an elliptic differential operator in divergence form subject to mixed boundary conditions. We show that $L$ possesses the Kato square root property, i.e. the domain of $L^\frac{1}{2}$ ... More Analogues of Entropy in Bi-Free Probability Theory: MicrostatesFeb 11 2019In this paper, we extend the notion of microstate free entropy to the bi-free setting. In particular, using the bi-free analogue of random matrices, microstate bi-free entropy is defined. Properties essential to an entropy theory are developed, such as ... More Analytic Functional Calculus in Quaternionic FrameworkFeb 11 2019Regarding quaternions as normal matrices, we first characterize the $2\times 2$ matrix-valued functions, defined on subsets of quaternions, whose values are quaternions. Then we investigate the regularity of quaternionic-valued functions, defined by the ... More The Positive Maximum Principle on Symmetric SpacesFeb 11 2019We investigate the Courr\`{e}ge theorem in the context of linear operators $A$ that satisfy the positive maximum principle on a space of continuous functions over a symmetric space. Applications are given to Feller--Markov processes. We also introduce ... More Operator algebras of higher rank numerical semigroupsFeb 10 2019A higher rank numerical semigroup is a positive cone whose seminormalization is isomorphic to the free abelian semigroup. The corresponding nonselfadjoint semigroup algebras are known to provide examples that answer Arveson's Dilation Problem to the negative. ... More Equivariant homologies for operator algebrasFeb 10 2019This is a survey of a variety of equivariant (co)homology theories for operator algebras. We briefly discuss a background on equivariant theories, such as equivariant $K$-theory and equivariant cyclic homology. As the main focus, we discuss a notion of ... More lacunary Walsh series in rearrangement invariant spacesFeb 10 2019We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series. Universal optimal configurations for the $p$-frame potentialsFeb 09 2019Given $d, N\geq 2$ and $p\in (0, \infty]$ we consider a family of functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all collections of $N$ unit-norm vectors in $\mathbb R^d$. For the special case $p=2$ and $p=\infty$, both the ... More $m_{n}$-Distributional chaos in Fr\' echet spacesFeb 09 2019The main aim of this paper is to introduce the concepts of $m_{n}$-distributional chaos and $\lambda$-distributional chaos for linear continuous operators and their sequences in Fr\' echet spaces ($\lambda \in (0,1]$), as well as their continuous analogues ... More Quasi-linear functionals on locally compact spacesFeb 09 2019This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested in quasi-linear ... More Generalization of Kuratowski problem in linear spacesFeb 08 2019In this short paper, Kuratowski problem will be investigated in vector space. The highest number of distinct sets that can be generated from one convex set in linear space by repeatedly applying algebraic closure and complement in any order is 8. Keywords: ... More Quantum Markov States on Cayley treesFeb 08 2019It is known that any locally faithful quantum Markov state (QMS) on one dimensional setting can be considered as a Gibbs state associated with Hamiltonian with commuting nearest-neighbor interactions. In our previous results, we have investigated quantum ... More Mixed operators on $L^p$-direct integralsFeb 08 2019The notion of decomposable operators acting between different $L^p$-direct integrals of Banach spaces is introduced. We show that those operators generalize the composition operator, in the sense that a mapping is replaced by a binary relation. The necessary ... More Superposition, reduction of multivariable problems, and approximationFeb 07 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... Moremath.FAmath.PRPrimary 47L60, 46N30, 46N50, 42C15, 65R10, 31C20, 62D05, 94A20,
39A12, Secondary 46N20, 22E70, 31A15, 58J65 Polynomial inequalities on the Hamming cubeFeb 06 2019Let $(X,\|\cdot\|_X)$ be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $f:\{-1,1\}^n\to X$ on the Hamming cube whose spectrum is bounded above or below. Our proofs ... More On a problem of PichoridesFeb 06 2019Let $S^{(\Lambda)}$ denote the classical Littlewood-Paley square function formed with respect to a lacunary sequence $\Lambda$ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator ... More On the Operator Equations $A^n=A^*A$Feb 06 2019Let $n\in\mathbb{N}$ and let $A$ be a closed linear operator (everywhere bounded or unbounded). In this paper, we study (among others) equations of the type $A^*A=A^n$ where $n\geq2$ and see when they yield $A=A^*$ (or a weaker class of operators). In ... More Hardy spaces of general Dirichlet series - a surveyFeb 06 2019The main purpose of this article is to survey on some key elements of a recent $\mathcal{H}_p$-theory of general Dirichlet series $\sum a_n e^{-\lambda_{n}s}$, which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series $\sum ... More Weighted Translation Semigroups: Multivariable Case-IFeb 05 2019M. Embry and A. Lambert initiated the study of a weighted translation semigroup {S_t}, with a view to explore a continuous analogue of a weighted shift operator. We continued the work, characterized some special types of semigroups and developed an analytic ... More On the geometry of random polytopesFeb 05 2019We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma=(\xi_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $\xi$, and set $X_1,...,X_N$ ... More Notes on bilinear multipliers on Orlicz spacesFeb 04 2019Let $\Phi_1 , \Phi_2 $ and $ \Phi_3$ be Young functions and let $L^{\Phi_1}(\mathbb{R})$, $L^{\Phi_2}(\mathbb{R})$ and $L^{\Phi_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(\xi,\eta)$ defined on $\mathbb{R}\times \mathbb{R}$ ... More On switching probability measures and questions of KardarasFeb 03 2019Let $\mathcal{K}$ be a convex bounded positive set in $\mathbb{L}^1(\mathbb{P})$. Kardaras [6] asked the following two questions: (1) If the relative $\mathbb{L}^0(\mathbb{P})$-topology is locally convex on $\mathcal{K}$, does there exist $\mathbb{Q}\sim ... More SSGP topologies on free groups of infinite rankFeb 03 2019We prove that every free group G with infinitely many generators admits a Hausdorff group topology T with the following property: for every T-open neighbourhood U of the identity of G, each element g in G can be represented as a product g=g_1 g_2 ... ... More Ergodic theorems in Banach ideals of compact operatorsFeb 02 2019Let $\mathcal H$ be a complex infinite-dimensional Hilbert space, and let $\mathcal B(\mathcal H)$ ($\mathcal K(\mathcal H)$) be the $C^*$-algebra of bounded (respectively, compact) linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a fully symmetric ... More On some spectral properties of pseudo-differential operators on TJan 31 2019Feb 07 2019In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb{T} := \mathbb{R}/ 2 \pi \mathbb{ Z}$. For symbols in the H\"ormander ... More On some spectral properties of pseudo-differential operators on TJan 31 2019Feb 14 2019In this paper we use Riesz spectral Theory and Gershgorin Theory to obtain explicit information concerning the spectrum of pseudo-differential operators defined on the unit circle $\mathbb{T} := \mathbb{R}/ 2 \pi \mathbb{ Z}$. For symbols in the H\"ormander ... More